The hyperbolic class of quadratic time-frequency representations. II. Subclasses, intersection with the affine and power classes, regularity, and unitarity

For pt.I see ibid., vol.41, p.3425-444 (1993). Part I introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFRs). The present paper defines and studies four subclasses of the HC: (1) The focalized-kernel subclass of the HC is related to a time-frequency concentration property of QTFRs. It is analogous to the localized-kernel subclass of the affine QTFR class. (2) The affine subclass of the HC (affine HC) consists of all HC QTFRs that satisfy the conventional time-shift covariance property. It forms the intersection of the HC with the affine QTFR class. (3) The power subclasses of the HC consist of all HC QTFRs that satisfy a “power time-shift” covariance property. They form the intersection of the HC with the recently introduced power classes. (4) The power-warp subclass of the HC consists of all HC QTFRs that satisfy a covariance to power-law frequency warpings. It is the HC counterpart of the shift-scale covariant subclass of Cohen´s class. All of these subclasses are characterized by 1D kernel functions. The affine HC is contained in both the localized kernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary Po-distribution by a convolution. We furthermore consider the properties of regularity and unitarity in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for least-squares signal synthesis and new relations for the Altes-Marinovich Q-distribution